Homomorphisms into simple ${\cal Z}$-stable $C^*$-algebras

Abstract

Let A and B be unital separable simple amenable C*-algebras which satisfy the Universal Coefficient Theorem. Suppose that A and B are Z-stable and are of rationally tracial rank no more than one. We prove the following: Suppose that φ, ψ : A → B are unital *monomorphisms. There exists a sequence of unitaries {un} ⊂ B such that lim n→∞ unφ(a)un = ψ(a) for all a ∈ A, if and only if [φ] = [ψ] in KL(A,B), φ] = ψ] and φ ‡ = ψ‡, where φ], ψ] : Aff(T(A)) → Aff(T(B)) and φ‡, ψ‡ : U(A)/CU(A) → U(B)/CU(B) are the induced maps (where T(A) and T(B) are the tracial state spaces of A and B, and CU(A) and CU(B) are the closures of the commutator subgroups of the unitary groups of A and B, respectively). We also show that this holds if A is a rationally AH-algebra which is not necessarily simple. Moreover, for any strictly positive unit-preserving κ ∈ KL(A,B), any continuous affine map λ : Aff(T(A))→ Aff(T(B)) and any continuous group homomorphism γ : U(A)/CU(A)→ U(B)/CU(B) which are compatible, we also show that there is a unital homomorphism φ : A→ B so that ([φ], φ], φ ‡) = (κ, λ, γ), at least in the case that K1(A) is a free group.